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Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuniss 3601 Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremssuni 3602 Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremunissi 3603 Subclass relationship for subclass union. Inference form of uniss 3601. (Contributed by David Moews, 1-May-2017.)

Theoremunissd 3604 Subclass relationship for subclass union. Deduction form of uniss 3601. (Contributed by David Moews, 1-May-2017.)

Theoremuni0b 3605 The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)

Theoremuni0c 3606* The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)

Theoremuni0 3607 The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)

Theoremelssuni 3608 An element of a class is a subclass of its union. Theorem 8.6 of [Quine] p. 54. Also the basis for Proposition 7.20 of [TakeutiZaring] p. 40. (Contributed by NM, 6-Jun-1994.)

Theoremunissel 3609 Condition turning a subclass relationship for union into an equality. (Contributed by NM, 18-Jul-2006.)

Theoremunissb 3610* Relationship involving membership, subset, and union. Exercise 5 of [Enderton] p. 26 and its converse. (Contributed by NM, 20-Sep-2003.)

Theoremuniss2 3611* A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)

Theoremunidif 3612* If the difference contains the largest members of , then the union of the difference is the union of . (Contributed by NM, 22-Mar-2004.)

Theoremssunieq 3613* Relationship implying union. (Contributed by NM, 10-Nov-1999.)

Theoremunimax 3614* Any member of a class is the largest of those members that it includes. (Contributed by NM, 13-Aug-2002.)

2.1.19  The intersection of a class

Syntaxcint 3615 Extend class notation to include the intersection of a class (read: 'intersect ').

Definitiondf-int 3616* Define the intersection of a class. Definition 7.35 of [TakeutiZaring] p. 44. For example, { { 1 , 3 } , { 1 , 8 } } = { 1 } . Compare this with the intersection of two classes, df-in 2924. (Contributed by NM, 18-Aug-1993.)

Theoremdfint2 3617* Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)

Theoreminteq 3618 Equality law for intersection. (Contributed by NM, 13-Sep-1999.)

Theoreminteqi 3619 Equality inference for class intersection. (Contributed by NM, 2-Sep-2003.)

Theoreminteqd 3620 Equality deduction for class intersection. (Contributed by NM, 2-Sep-2003.)

Theoremelint 3621* Membership in class intersection. (Contributed by NM, 21-May-1994.)

Theoremelint2 3622* Membership in class intersection. (Contributed by NM, 14-Oct-1999.)

Theoremelintg 3623* Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)

Theoremelinti 3624 Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremnfint 3625 Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Theoremelintab 3626* Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)

Theoremelintrab 3627* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)

Theoremelintrabg 3628* Membership in the intersection of a class abstraction. (Contributed by NM, 17-Feb-2007.)

Theoremint0 3629 The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.)

Theoremintss1 3630 An element of a class includes the intersection of the class. Exercise 4 of [TakeutiZaring] p. 44 (with correction), generalized to classes. (Contributed by NM, 18-Nov-1995.)

Theoremssint 3631* Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)

Theoremssintab 3632* Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremssintub 3633* Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.)

Theoremssmin 3634* Subclass of the minimum value of class of supersets. (Contributed by NM, 10-Aug-2006.)

Theoremintmin 3635* Any member of a class is the smallest of those members that include it. (Contributed by NM, 13-Aug-2002.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Theoremintss 3636 Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)

Theoremintssunim 3637* The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)

Theoremssintrab 3638* Subclass of the intersection of a restricted class builder. (Contributed by NM, 30-Jan-2015.)

Theoremintssuni2m 3639* Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)

Theoremintminss 3640* Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)

Theoremintmin2 3641* Any set is the smallest of all sets that include it. (Contributed by NM, 20-Sep-2003.)

Theoremintmin3 3642* Under subset ordering, the intersection of a class abstraction is less than or equal to any of its members. (Contributed by NM, 3-Jul-2005.)

Theoremintmin4 3643* Elimination of a conjunct in a class intersection. (Contributed by NM, 31-Jul-2006.)

Theoremintab 3644* The intersection of a special case of a class abstraction. may be free in and , which can be thought of a and . (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremint0el 3645 The intersection of a class containing the empty set is empty. (Contributed by NM, 24-Apr-2004.)

Theoremintun 3646 The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)

Theoremintpr 3647 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.)

Theoremintprg 3648 The intersection of a pair is the intersection of its members. Closed form of intpr 3647. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.)

Theoremintsng 3649 Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Theoremintsn 3650 The intersection of a singleton is its member. Theorem 70 of [Suppes] p. 41. (Contributed by NM, 29-Sep-2002.)

Theoremuniintsnr 3651* The union and intersection of a singleton are equal. See also eusn 3444. (Contributed by Jim Kingdon, 14-Aug-2018.)

Theoremuniintabim 3652 The union and the intersection of a class abstraction are equal if there is a unique satisfying value of . (Contributed by Jim Kingdon, 14-Aug-2018.)

Theoremintunsn 3653 Theorem joining a singleton to an intersection. (Contributed by NM, 29-Sep-2002.)

Theoremrint0 3654 Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremelrint 3655* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Theoremelrint2 3656* Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)

2.1.20  Indexed union and intersection

Syntaxciun 3657 Extend class notation to include indexed union. Note: Historically (prior to 21-Oct-2005), set.mm used the notation , with the same union symbol as cuni 3580. While that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses as distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.

Syntaxciin 3658 Extend class notation to include indexed intersection. Note: Historically (prior to 21-Oct-2005), set.mm used the notation , with the same intersection symbol as cint 3615. Although that syntax was unambiguous, it did not allow for LALR parsing of the syntax constructions in set.mm. The new syntax uses a distinguished symbol instead of and does allow LALR parsing. Thanks to Peter Backes for suggesting this change.

Definitiondf-iun 3659* Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, is independent of (although this is not required by the definition), and depends on i.e. can be read informally as . We call the index, the index set, and the indexed set. In most books, is written as a subscript or underneath a union symbol . We use a special union symbol to make it easier to distinguish from plain class union. In many theorems, you will see that and are in the same distinct variable group (meaning cannot depend on ) and that and do not share a distinct variable group (meaning that can be thought of as i.e. can be substituted with a class expression containing ). An alternate definition tying indexed union to ordinary union is dfiun2 3691. Theorem uniiun 3710 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)

Definitiondf-iin 3660* Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3659. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3692. Theorem intiin 3711 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)

Theoremeliun 3661* Membership in indexed union. (Contributed by NM, 3-Sep-2003.)

Theoremeliin 3662* Membership in indexed intersection. (Contributed by NM, 3-Sep-2003.)

Theoremiuncom 3663* Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)

Theoremiuncom4 3664 Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)

Theoremiunconstm 3665* Indexed union of a constant class, i.e. where does not depend on . (Contributed by Jim Kingdon, 15-Aug-2018.)

Theoremiinconstm 3666* Indexed intersection of a constant class, i.e. where does not depend on . (Contributed by Jim Kingdon, 19-Dec-2018.)

Theoremiuniin 3667* Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiunss1 3668* Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiinss1 3669* Subclass theorem for indexed union. (Contributed by NM, 24-Jan-2012.)

Theoremiuneq1 3670* Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)

Theoremiineq1 3671* Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)

Theoremss2iun 3672 Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2 3673 Equality theorem for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2 3674 Equality theorem for indexed intersection. (Contributed by NM, 22-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremiuneq2i 3675 Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2i 3676 Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)

Theoremiineq2d 3677 Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)

Theoremiuneq2dv 3678* Equality deduction for indexed union. (Contributed by NM, 3-Aug-2004.)

Theoremiineq2dv 3679* Equality deduction for indexed intersection. (Contributed by NM, 3-Aug-2004.)

Theoremiuneq1d 3680* Equality theorem for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq12d 3681* Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.)

Theoremiuneq2d 3682* Equality deduction for indexed union. (Contributed by Drahflow, 22-Oct-2015.)

Theoremnfiunxy 3683* Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiinxy 3684* Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiunya 3685* Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiinya 3686* Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)

Theoremnfiu1 3687 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.)

Theoremnfii1 3688 Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)

Theoremdfiun2g 3689* Alternate definition of indexed union when is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremdfiin2g 3690* Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)

Theoremdfiun2 3691* Alternate definition of indexed union when is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)

Theoremdfiin2 3692* Alternate definition of indexed intersection when is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremdfiunv2 3693* Define double indexed union. (Contributed by FL, 6-Nov-2013.)

Theoremcbviun 3694* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.)

Theoremcbviin 3695* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)

Theoremcbviunv 3696* Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.)

Theoremcbviinv 3697* Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.)

Theoremiunss 3698* Subset theorem for an indexed union. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun 3699* Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Theoremssiun2 3700 Identity law for subset of an indexed union. (Contributed by NM, 12-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

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